I'm not entirely sure of what you are asking, but note that Erdos and Lehner proved <a href="https://projecteuclid.org/journals/duke-mathematical-journal/volume-8/issue-2/The-distribution-of-the-number-of-summands-in-the-partitions/10.1215/S0012-7094-41-00826-8.short">here</a> that
$$p(n,m)\sim \frac{n^{m-1}}{m!(m-1)!}$$ holds for $m=o(n^{1/3})$. In generality for any finite set $A$, with $|A|=m$ and $p(n,A)$ denoting the number of partitions of $n$ with parts from $A$, one has
$$p(n,A)=\frac{1}{\prod_{a\in A}a}\frac{n^{m-1}}{(m-1)!}+O(n^{m-2}).$$

Such estimations can be deduced from the generating function of $p$ by using methods that are described in many books, for example "Analytic Combinatorics" by Flajolet and Sedgewick.